{
  "abstract": "For a family $\\mathcal F\\subseteq\\binom{[n]}k$ and $R\\in\\binom{[n]}r$, let $d_{\\mathcal F}(R)=|\\{F\\in\\mathcal F:R\\subseteq F\\}|$ and $\\ell_{r,p}(\\mathcal F)=\\sum_{R\\in\\binom{[n]}r}d_{\\mathcal F}(R)^p$; write $co_p(\\mathcal F)=\\ell_{k-1,p}(\\mathcal F)$ for the codegree power sum. We introduce a method that uses convexity to extend sharp bounds for degree sums and sums of squared degrees to real powers, while retaining all equality cases. The method bounds $x^p$ by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with $x^p$ at the degrees of the proposed extremal family, so the argument requires no bounds for sums of higher powers. For families with bounded matching number, we instead use a bound for $\\sum_R\\max\\{d_{\\mathcal F}(R)-s,0\\}$ together with the degree sum. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\\geq2$ in the sharp range $n\\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the quadratic theorem of Wu and Zhang to real exponents and answers a problem of Zhou and Yuan throughout the sharp Erdős--Ko--Rado range. Second, among intersecting families with $n\\geq2k$, a full star maximizes $\\ell_{r,p}$ for every $1\\leq r\\leq k-1$ and real $p\\geq2$, again with all equality cases determined. Third, if $ν(\\mathcal F)\\leq s$ and $n\\geq(2s+1)k-s$, then for every real $p\\geq1$, $co_p(\\mathcal F)$ is uniquely maximized, up to permutation, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or assumptions that $n$ is sufficiently large with an explicit linear range valid for arbitrary $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.28616",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengyu Cao"
    },
    {
      "name": "Mei Lu"
    },
    {
      "name": "Haixiang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b0383kh9k7c3c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.28616",
  "fulltext_url": null,
  "id": "b0383kh9k7c3c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.28616",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7171978345",
  "pdf_url": "https://arxiv.org/pdf/2607.28616",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b0383kh9k7c3c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2607.28616",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.28616"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Degree Power Sums in Extremal Set Systems",
  "title_text": "Degree Power Sums in Extremal Set Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/b0383kh9k7c3c",
  "venue": "arXiv",
  "versions": []
}